The false statement among the ones provided is:

"The greater the improvement in model fit from moving from the simple to complex model the lower the F-statistic."

Explanation:

• F-statistic and Model Improvement: The F-statistic in ANOVA and regression measures how much the regression model improves the fit of the data compared to a baseline model (often a simple model, like the intercept-only model). A greater improvement in the model fit (i.e., a more significant change from the simple to complex model) results in a larger F-statistic, not a smaller one. The larger the difference between the variance explained by the model and the variance left unexplained (error variance), the higher the F-statistic.

The other statements are true:

1. ANOVA splits total variance: ANOVA decomposes the total variance into components explained by the regression model (the explained variance) and the residual variance (the unexplained error).

2. ANOVA as a joint test of significance: ANOVA tests the null hypothesis that all model coefficients (except the intercept) are equal to zero. This is a joint test of the significance of the model predictors as a whole.

3. Two predictor regression and ANOVA null: In a regression with two predictors, the null hypothesis for ANOVA is that the coefficients for both predictors are zero, i.e., $H_0: \beta_1 = \beta_2 = 0$.

Would you like more details on how the F-statistic works in regression models?

Here are some related questions you might find useful:

1. What is the formula for the F-statistic in regression analysis?
2. How do you interpret the p-value associated with the F-statistic in ANOVA?
3. What is the relationship between R-squared and the F-statistic in regression?
4. How can ANOVA be used to compare nested models?
5. What are the assumptions underlying ANOVA in regression analysis?

Tip: Always check the degrees of freedom when interpreting the F-statistic, as it affects the critical value you compare it to.